The asymptotic matching method and the linear boundary value problem.

Begin by solving the problem exactly.

The indicial equation:

ε = 0.05

sol = Solve[ε*m^2 + m + 1 == 0, m]

mplus = m /. sol[[2]]

mminus = m /. sol[[1]]

yexact[
        t_] = (Exp[mplus*t] - Exp[mminus*t])/(Exp[mplus] - 
            Exp[mminus])

Plot[yexact[t], {t, 0, 1}]

Now consider the inner and outer approximations:

yinner[t_] = E*(1 - Exp[-t/ε])

youter[t_] = Exp[1 - t]

Plot[{yinner[t], youter[t], yexact[t]}, {t, 0, 1}]

Finally, we consider the "composite" approximation.

ycomposite[t_] = yinner[t] + youter[t] - E

Plot[{yexact[t], ycomposite[t]}, {t, 0, 1}]